Accurate Interaction Energies by Spin Component
Scaled M¨oller-Plesset Second Order Perturbation
Theory Calculations with Optimized Basis Sets
(scs-mp2
mod
): development and application to
aromatic heterocycles.
Ivo Cacelli,
a,b,∗Filippo Lipparini,
aLeandro Greff da Silveira,
cMatheus Jacobs,
d,ePaolo Roberto Livotto
cand Giacomo Prampolini
baDipartimento di Chimica e Chimica Industriale, Universit`a di Pisa,
Via G. Moruzzi 13, I-56124 Pisa, Italy
bIstituto di Chimica dei Composti OrganoMetallici (ICCOM-CNR),
Area della Ricerca, via G. Moruzzi 1, I-56124 Pisa, Italy
cInstituto de Qu´ımica, Universidade Federal do Rio Grande do Sul,
Avenida Bento Gon¸calves 9500, CEP 91501-970 Porto Alegre, Brazil
dInstitut f¨ur Physik, Humboldt-Universit¨at zu Berlin,
Newtonstr. 15, 12489, Berlin, Germany
eIRIS Adelrshof, Humboldt-Universit¨at zu Berlin,
Zum Großen Windkanal 6, 12489, Berlin, Germany
Abstract
The Spin Component Scaled (SCS) MP2 method using a reduced and optimized basis set (scs-mp2mod) is employed to compute the interaction energies of nine homo-dimers, formed by aromatic hetero-cyclic molecules (pyrrole, furan, thiophene, oxazole, isoxazole, pyridine, pyridazine, pyrimidine and pyrazine). The coefficients of the same-spin and opposite-spin correlation energies and the GTO polarization exponents of the 6-31G** basis set are simultaneously optimized in order to mini-mize the energy differences with respect to the CCSD(T) reference interaction en-ergies, extrapolated to complete basis set. It is demonstrated that the optimization of the spin scale factors leads to a noticeable improvement of the accuracy with a root mean square deviation less than 0.1 kcal/mol and a largest unsigned deviation smaller than 0.25 kcal/mol. The pyrrole dimer provides an exception, with slightly higher deviation from the reference data. Given the high benefit in terms of compu-tational time with respect to the CCSD(T) technique and the small loss of accuracy, the scs-mp2mod method appears to be particularly indicated for extensive sampling of intermolecular potential energy surfaces at a quantum mechanical level. Within this framework, the results of exponents and scaling factors optimization for the whole set of molecules are again accurate, showing a good level of transferability to this class of molecules.
1
Introduction
Intermolecular interactions are responsible for the existence of any type of condensed matter, and therefore they have received much attention in the fields of physics,
chem-istry, biology, nano-sciences and engineering.1–4 An accurate determination of the forces
taking place when two or more molecules approach each other, allows in principle for a rationalization of a number of experimental observations, including bulk phase properties. This task is however rather difficult, as the most reliable methods, as those based on the Coupled Cluster (CC) Ansatz, are limited to medium-sized molecules and to a moderate number of points of the interaction potential energy surface (IPES), due to their high computational cost. For this reason a number of approaches based on cheaper methods
exploiting both wave function5–11 and DFT families12–16were developed in recent years
(see also Ref. [17] for an excellent review). In this paper we give a contribution to this research line with the aim to improve a general method to compute bulk phase properties, as detailed in the following.
Several years ago we developed in our group a protocol to calculate the bulk properties of pure substances by Molecular Dynamics (MD) and/or Monte Carlo ( MC) simulation methods using a Force Field (FF) completely derived by ab initio quantum mechanical
calculations.18–25The parameters entering the FF were derived separately for the intra- and
inter-molecular interactions respectively using the Joyce26–28and Picky22–24 protocols.
The former task can be accomplished with a negligible computational cost, as it only requires the equilibrium geometry and Hessian as derived for the isolated target molecules, which can be computed with DFT. On the contrary, a robust determination of the inter-molecular FF parameters requires a large number of reference calculations at hundreds of points on the IPES. This step is very time consuming and a manageable although accurate quantum mechanical method is mandatory for the feasibility of the whole protocol. In a
first attempt performed on the benzene molecule18 we have used the MP2 method with
a small 6-31G* basis set, by setting the exponents of the polarization GTO functions of
gained in a challenging calculation of nematogenic molecule in its condensed phases20and
of IPES portions of the quinhydrone dimer and eumelanin building blocks.32, 33 Since
then, the name mp2mod was proposed to indicate MP2 calculations with modified basis
sets, suitable for intermolecular calculations. In a very recent work,25, 34 we applied the
same method to a variety of homo-dimers, formed by hetero-cyclic aromatic molecules. The GTO exponents of the polarization functions (d for N,C,O and p for H atoms) of
a 6-31G** basis set were optimized by fitting the mp2mod results to CCSD(T) energies,
extrapolated at in the complete basis set (CBS) limit, at a number of dimer geometries for several aromatic hetero-cycles, namely pyrrole, furan, thiophene, oxazole, isoxazole,
pyridine, pyridazine, pyrimidine and pyrazine.34 The resulting system specific basis sets
were then used in mp2mod calculations for about 500 points of the IPES for each system,
sorted with the Picky protocol. The FF parameters were then determined exploiting this
extended mp2mod database and successively used in MD to compute the bulk properties.25
In this paper, we calculate the intermolecular energies for several homo- and
hetero-dimers of the aforementioned species, using again the mp2mod/6-31G** model, but refining
it with the use of Spin Component Scaled (SCS) correlation energies. By this method, first
proposed by Grimme,5 the correlation energy as obtained by standard MP2 calculations
is recomputed by weighting the same-spin (SS) and opposite-spin (OS) components with two independent scaling factors
Ecorr(SCS − M P 2) = CSSESS + COSEOS
Although this procedure was designed to improve the MP2 results on isolated molecules, SCS-MP2 was successively considered in intermolecular calculations, in several attempts
aimed to find the optimal scaling coefficients for binding energy calculations.6, 17, 35–38 The
main conclusions of these studies may be summarized as follows.
1. By exploiting the advantage of tuning two independent parameters, SCS-MP2 per-forms better than MP2, as the root mean square deviation from accurate data
2. The optimal scaling factors are very different from those found for single molecule calculations and, specifically, the same-spin component is enhanced whereas the opposite-spin component is reduced, and in some cases may become even negative.
For specific systems only the same-spin component was used.38
In this paper we study the aforementioned hetero-aromatic dimers in several conforma-tions with the aim to improve the previous results by optimizing both the exponents of the polarization GTO functions and the MP2 spin scaling factors. The parameters for SCS-MP2 were first optimized specifically for each considered homo-dimer, aiming to reach the best possible accuracy with respect to reference data. In a attempt to check the transferability of the exponents and the spin scaling factors further calculations were performed. In the first one all the five membered ring molecules were constrained to take the same parameters; in the second one the same was done for the six membered ring systems and finally, all molecules here considered takes the same exponents and scaling factors through a fitting using all the reference data.
2
Method and Computational Details
The same-spin (SS) and opposite-spin (OS) components of the MP2 correlation energy of a closed shell system have the following expression
ESS = − X ijab hij|abi(hij|abi − hij|bai) ǫa+ ǫb− ǫi− ǫj (1) EOS = − X ijab |hij|abi|2 ǫa+ ǫb− ǫi− ǫj (2) where i, j and a, b run on the occupied and virtual spatial orbitals, respectively. The two-electron integrals are always written in the Dirac notation. The Spin Component Scaled (SCS) correlation energy is a linear combination of both components through the scaling factors
The optimization of both the exponents of the GTO polarization functions of the 6-31G**
basis set and the scaling factors CSS and COS, was performed by fitting the resulting
interaction energies with a reference database determined by CCSD(T)/CBS results, often
defined as quantum chemical gold standard.39
In most of its current implementations,40the CCSD(T)/CBS interaction energy can be
recovered from the MP2/CBS limit (∆EM P 2
CBS), corrected with the difference (∆CCSD(T ))
between the MP2 and CCSD(T) interaction energy, computed with a smaller X basis set: ∆ECBSCCSD(T ) = ∆ECBSM P 2+ ∆EXCCSD(T )− ∆EXM P 2 (4) Although this procedure has been extensively used by several groups (see references [39–
45], just to cite a few), the best route to correctly estimate both ∆EM P 2
CBS and ∆CCSD(T )
values has not yet been uniquely assessed. In consideration of the large number of CCSD(T) calculations required, in this paper the MP2 interaction energy at CBS was
estimated by the Halkier extrapolation scheme46 employing the cc-pVDZ and
aug-cc-pVTZ basis sets, whereas the ∆CCSD(T ) correction was computed exploiting the
aug-cc-pVDZ basis set. Indeed, in previous work,34 it was found, at least for the benzene
dimer, that this choice did not introduce appreciable differences ( 0.01 kcal/mol) with
re-spect to more refined schemes,44, 45which employed aug-cc-pVTZ and aug-cc-pVQZ basis
set in the Halkier extrapolation, and heavy-aug-cc-pVDZ/heavy-aug-cc-pVTZ two-point extrapolation basis set for ∆CCSD(T ).
The interaction energy ∆E of a dimer A...B is defined as
∆E = EAB − EA− EB (5)
where the E’s are the absolute energies of the super-molecule or of the single molecules for any type of calculation. In all CCSD(T) and MP2 calculations the basis set superposition
error (BSSE) was taken into account by the standard Counterpoise (CP) correction47 so
that all energies entering the above expression are computed with the basis set of the AB molecule at the considered geometry. The complete interaction energy of a SCS-MP2 calculation is evaluated by the Hartree-Fock, same-spin and opposite-spin interaction
energies
∆ESCS−MP 2 = ∆EHF + CSS∆ESS + COS∆EOS (6)
The equilibrium geometry of each isolated monomer was taken from Ref. [34], where
it was obtained by geometry optimization at DFT level, using the B3LYP functional joined with the Dunning’s correlation consistent cc-pVTZ basis set. All the calculations concerning monomers and homo-dimers were performed with the Gaussian09 suite of
programs.48 All the calculations concerning hetero-dimers were performed with a locally
modified version of the CFOUR suite of programs.49, 50 Geometry optimizations for the
hetero-dimers at the scs-mp2mod level of theory were performed with CFOUR using
analytical gradients.51
In system-specific parameterizations, the GTO exponents and the scaling factors were optimized separately for each system and the reference dimer database is formed by 4 or 5 curves, corresponding to the most common arrangements for the considered homo-dimer reported in literature: face-to-face (both parallel and anti-parallel), T-shaped, parallel displaced and other system specific arrangements which span attractive regions of the IPES. A complete collection of the dimer geometries is shown in the Supporting Informa-tion, along with the detailed instructions to get the geometrical arrangements from the translational vectors and the Euler angles defined therein in Figure A. Three additional optimizations were attempted, with the aim to obtain more general (yet probably less ac-curate) parameters, possibly suitable for interaction energy SCS-MP2 calculations in any pair of hetero-cycles. Namely, two parameterizations were performed with respect to the database built with all the CCSD(T)/CBS interaction energies computed for either the five- or six-membered ring systems, whereas a final one was performed over the database comprising the whole set of computed CCSD(T)/CBS data. In all cases, all molecules pertaining to the chosen training set were equipped with the same exponents and
scal-ing factors, thus offerscal-ing in principle the possibility to use SCS-mp2mod for any pair of
five-membered, six-membered or generic hetero-cyclic aromatic molecules, respectively. For each parameterization, the GTO exponents and scaling factors (herein also referred
to as the parameters) were optimized by minimizing the Root Mean Square Deviation
(RMSD) of the mp2mod energies with respect to the CCSD(T)/CBS ones,
RMSD(α, CSS, COS) =
Np X
i=1
[EiM P 2(α, CSS, COS) − Ei(CCSD(T )/CBS]]2/Np (7)
where Np is the number of geometrical arrangements of the system (or systems) under
scrutiny and α indicates the collection of exponents of the polarization GTOs. The best-fit parameters were determined using the ”in house” program Exopt. At each optimization
cycle one exponent is varied, the new mp2mod energy components are computed for all
dimer arrangements and the scaling factors are optimized to give the new value of the RMSD. The C values were constrained to be positive or null, whereas the GTO exponents
were confined in the 0.05 - 1.5 Bohr−2 range.
3
Results
3.1
System-specific SCS-MP2
modIn this section we will present the results of the optimization of both basis set exponents and spin component scaling factor carried out separately for each molecule, i.e. using as reference CCSD(T)/CBS interaction energies of a specific homo-dimer in different arrangements. For each considered species, different classes of homo-dimer geometrical dispositions were hence devised, by displacing one monomer with respect to the other (see Supporting Information for a detailed definition). First, a preliminary interaction energy
curve was obtained at mp2mod level34 for each arrangement, by displacing one monomer
along the translation vector ~R. Next, a number of geometries was selected along the curve,
and their interaction energy computed at CCSD(T)/CBS level. The latter values were then used as a reference for the optimization procedure, performed separately for each species. An example of the resulting curves is displayed for pyridine in Figure 1, whereas all the remaining curves are shown in Figures B-J in the Supporting Information.
The main results of the present work are graphically summarized in Figure 2 and
ref-erence data clearly appears from the bottom panel of Figure 2, where the correlation plot between the two set of energies is displayed for all compounds. A more detailed analysis, taking in consideration the different geometrical classes for each dimer, is displayed in the Supporting Information (Figure K). In general, as evidenced from the top panel of
Figure 2, all scs-mp2mod energies are less than 0.4 kcal/mol different from the reference
CCSD(T)/CBS counterparts.
On the same foot, the statistical analysis reported in Table 1 confirms the close
agree-ment between the representation offered by the current scs-mp2mod model and the
ref-erence CCSD(T)/CBS database. In fact, the noticeable improvement provided by tuning the SCS scaling factors is apparent: all the usual statistical descriptors show a markedly reduced error, with a particular improvement of the LUD, which on average becomes
1/3 of that given by the standard mp2mod scaling factors. As already found in previous
Figure 1: CCSD(T)/CBS and scs-mp2mod
interaction energies computed for the pyridine dimer ar-rangements shown in the insets.
papers,35, 36 this means that the two scaling factors are significant parameters that allow
for better flexibility and thus an improved agreement of the scs-mp2mod energy to the
CCSD(T)/CBS reference.
The ratio R = ∆ESS/∆EOS averaged along the curves of the several arrangement
classes, reported in Table 2, indicates a dependence on the relative orientation of the two molecules. Although it is hard to find a well defined rationale, it seems that the most relevant geometrical feature affecting R is given by the closest atoms in the dimer belonging to different monomers. The main conclusion is that in the cases where one
or two H atoms of one monomer approach the π cloud of the other one, ∆ESS becomes
larger than ∆EOS and R reaches values up to 1.3. In all other cases R is smaller than one.
Moreover, the ratio is almost constant along the abscissa of the single curves, showing a
-6 -4 -2 0 2 ∆ECCSD(T)/CBS (kcal/mol) -6 -4 -2 0 2 ∆ E SCS-MP2 (kcal/mol) -6 -4 -2 0 2 -6 -4 -2 0 2 Pyrrole Furan Thiophene Oxazole Isoxazole Pyridine Pyridazine Pyrimidine Pyrazine -1.5 -1 -0.5 0 -1.5 -1 -0.5 0 -0.4 -0.2 0 0.2 0.4 ∆ (kcal/mol)
Figure 2: Top: Signed error (∆ = ∆ESCS−M P2- ∆ECCSD(T )) (kcal/mol) between scs-mp2mod
com-puted values and CCSD(T)/CBS reference energies. Bottom: Correlation plot between scs-mp2mod
en-ergies and reference CCSD(T)/CBS data for the investigated aromatic hetero-cycles. The correlation of the [-2 – 0 ] kcal/mol range is evidenced in the inset.
NO-SCS SCS
N.pt RMSD MAD LUD RMSD MAD LUD
Pyrrole 23 0.410 0.230 0.980 0.141 0.101 0.313 Furan 24 0.198 0.142 0.601 0.061 0.052 0.125 Thiophene 26 0.204 0.129 0.704 0.085 0.068 0.198 Oxazole 26 0.229 0.163 0.642 0.101 0.079 0.254 Isoxazole 28 0.428 0.332 1.120 0.069 0.059 0.153 Pyridine 25 0.216 0.154 0.658 0.104 0.083 0.235 Pyridazine 26 0.237 0.159 0.716 0.098 0.077 0.262 Pyrimidine 27 0.250 0.168 0.862 0.102 0.080 0.220 Pyrazine 26 0.285 0.215 0.591 0 .082 0.065 0.206
Table 1: Results of the mp2mod
GTO exponents and scaling factors from individual optimization in terms of: Root Mean Square Deviation (RMSD), Mean Absolute Deviation (MAD) and Largest Unsigned Deviation (LUD) with respect to interaction energies computed at CCSD(T)/CBS level. The mp2mod
basis set is the modified 6-31G**. All energies are in kcal/mol.
dimer geometrical arrangement R = ∆ESS/∆EOS
parallel 0.7 - 0.9
parallel + lateral displacement 0.9 - 1.1
T-shaped (H pointing towards π cloud) 1.1 - 1.3
T-shaped (N,O,S pointing towards π cloud) 0.9 - 0-95
Table 2: Mean value of the R = ∆ESS/∆EOSalong the several geometrical arrangement considered in
the present work.
marked dependence on the relative orientation, rather than on the distance. By looking at equations (1) and (2), the difference between the two spin component interaction energies is roughly given by ∆EOS− ∆ESS ≃ A X ia B X jb hij|abihij|bai ǫa+ ǫb− ǫi− ǫj (8) where A and B indicates the set of orbitals of the two monomers. This formula may be a reasonable approximation in the hypothesis that the molecular orbitals are strongly localized on one monomer and that they resemble those of the single molecule. This approximate expression includes a number of terms with opposite sign and is rather difficult to be exploited in order to rationalize the observed R values.
In general, the energy residues are rather equally distributed along the dimer arrange-ments, meaning that different dimer classes are represented with the same quality by
scs-mp2mod, as also evident from a visual inspection of Figure 2. With few exceptions,
the short distance repulsive points are those showing the largest differences with respect to CCSD(T)/CBS energies. However, for such points the correlation energy is rather large and the extrapolation scheme might not be fully applicable in such a regime, making the
reference energies less reliable. The least accurate performances of scs-mp2mod are
ob-served for pyrrole, in all the three statistical indicators. As this system is able to give rise to one or two hydrogen bonds and arrangements that allow such an interaction were included in the database, pyrrole dimers were further investigated. Concretely, the GTO p exponent of the H atom bonded to Nitrogen and thus able to participate in hydrogen bonds was optimized as a further, independent parameter. The results were not encour-aging as the RMSD goes from 0.141 to 0.136 and the LUD goes from 0.313 to 0.300 kcal/mol. Therefore, the proposed strategy for improving the results is ineffective and the origin of the unusual higher deviations from CCSD(T)/CBS found for pyrrole with respect to the other systems seems to call for further investigations.
Nonetheless, despite the slightly worse performance registered for the pyrrole dimer,
it is evident from the top panel of Figure 2 that the scs-mp2mod procedure is able
to deliver a very accurate estimate (< 0.4 kcal/mol) of the interaction energies of the investigated dimers in a wide range of values (from -7 to +1 kcal/mol), as well as a balanced description of the relative stability of different arrangements. A further point of strength, which becomes crucial in an extensive sampling of dimer IPESs, consists in the computational benefit with respect to high level techniques. Indeed, as it appears from Table 3, the cost in terms of CPU time is decreased by three orders of magnitude with respect to a corresponding CCSD(T)/CBS calculation, thus allowing for the computation of hundreds of IPES points.
The optimal exponents and scaling factors for scs-mp2mod are reported in Table 4
separately for each investigated species. When SCS is not activated (left data in Table
Method basis set CPU time (m)
five membered rings six membered rings
MP2 aug-cc-pVDZ 45 90
MP2 aug-cc-pVTZ 1200 1700
CCSD(T) aug-cc-pVDZ 4100 7800
CCSD(T) CBS 5300 9600
scs-mp2mod 6-31G** 5 10
Table 3: Average CPU times on a single Xeon(R) 2.6 GHz processor employed in the calculation of
dimer interaction energies with different methods.
NO-SCS SCS αC αN αO/S αH αC αN αO/S αH CSS COS Pyrrole 0.244 0.364 0.546 0.216 0.455 0.946 1.259 0.941 Furan 0.217 0.326 0.187 0.598 0.195 0.399 1.874 0.633 Thiophene 0.230 0.218 0.341 0.263 0.231 1.500 1.515 0.639 Oxazole 0.247 0.266 0.368 0.176 0.690 0.562 0.405 0.588 1.771 0.692 Isoxazole 0.207 0.312 0.350 0.324 0.904 0.645 0.237 1.500 2.570 0.157 Pyridine 0.366 0.184 0.206 1.216 0.205 0.593 2.629 0.000 Pyridazine 0.482 0.266 0.207 1.240 0.439 0.184 2.161 0.180 Pyrimidine 0.415 0.301 0.219 0.903 0.653 0.328 2.444 0.000 Pyrazine 0.465 0.330 0.180 1.101 0.731 0.175 1.945 0.420
Table 4: Orbital exponents of the polarization GTOs and scaling factors of the mp2mod
optimization of the 6-31** basis set. All energies are in kcal/mol.
noticed that the parameters corresponding to the GTO exponents show a certain degree of redundancy, so that similar values for RM SD in equation (7) may sometimes be found at different values of the parameters, i.e. there might be some local minima rather close to each other. This implies that the specific GTO exponents have no particular meaning when considered independently of each other: it is only the full set of exponents that defines the 6-31G** basis set as a whole that makes it suitable for computing molecular
interactions with the scs-mp2mod scheme.
A further interesting result of the present work concerns with the scaling factors. In all cases the SS coefficient is higher than the OS one, but the behavior is different for the two sets of dimers formed by five member rings and six member rings. With the
exception of isoxazole, in the first group CSS is always in the 1.2 - 1.9 range and COS
takes values greater than 0.6, whereas in the second group CSS is even more dominant
with values close or greater than 2 whereas COS is rather small and would become even
negative after removal of the imposed constraint to be positive or null. For all systems the sum of the scaling factors is in the 2.1 - 2.7 range. Apart from some specificities connected both with the type of dimer and with the geometrical arrangement, a conclusion of the
present study is that in all cases CSS is the most relevant coefficient. This finding is
in line with practically all the authors who studied the intermolecular interactions by
the SCS-MP2 method. In particular, Antony and Grimme35 compared results given by
several scaling protocols of MP2 for the S22 data set, and found that the SCS-MP237, 38
(COS = 0 and CSS = 1.76) is the best performing one. The present results are also in
line with the conclusions of Distasio and Head-Gordon36who found the optimal SCS-MP2
scaling factors to be CSS = 1.29 and COS = 0.40 for the S22 data set with CCSD(T)/CBS
reference data. In the case of cc-pVTZ basis set the same authors found that CSS = 1.75
and COS = 0.17 are the best scaling factors, that are similar to the optimal values found
in the present work. The systematic higher contribution of ∆ESS to the intermolecular
energies at SCS-MP2 level is rather difficult to be rationalized. The considerations of
Grimme et al.13 concerning a connection of E
SS and EOS with the long-range and
short-range correlation effects, respectively, are stimulating and can furnish a sort of rationale, although they seem more adequate for intra-molecular correlation energies. In fact the dispersion energy (8) originates from simultaneous excitations of the two monomers and the short-range long-range distinction appears to be rather weak. For our purposes, in line with the empirical motivations of the SCS strategy, this systematic higher contribution
of ∆ESS can be considered for the moment a fortuitous feature which allows to greatly
improve the accuracy of the intermolecular energies with no additional computational cost.
One more comment on the results of Table 4 concerns with the high values of the sum of the scaling factors, joined with the general increase of the GTO exponents when using SCS. The optimal GTO exponents without using SCS are rather low for C,N,O
atoms and higher for the p functions of the Hydrogen atoms. The former are in line with the first attempt to evaluate the intermolecular energy of dispersion driven complexes by
MP2 with modified basis sets,18, 22, 29–31 where the GTO exponent of the d function of
the Carbon atoms was set to 0.25, below the one employed in standard single molecule calculations. On the contrary, when optimized for SCS-MP2, the GTO exponents tend in general to increase and in some cases they reach values comparable with those optimized for single molecule calculations. High exponents are suitable to describe the dynamic electronic correlation but can not correctly represent the polarization induced by the second molecule and give a small contribution to the molecular polarizability, which is a crucial ingredient in the evaluation of the dispersion energy. Therefore such exponent rise corresponds to a decrease of the MP2 interaction energies for both the same-spin and spin components. For example, the mean value of the same-spin and opposite-spin interaction energies of Pyridazine are -2.33 and -2.58 kcal/mol with the NO-SCS basis set and -2.13 and -2.37 kcal/mol for the SCS basis set, respectively. Thus it is apparent that in the SCS-MP2 calculations the interaction energy without scaling (i.e.
using CSS = COS = 1) would be underestimated. As a consequence, the scaling factors
need to be increased in order to match the reference values. In turn this implies that their sum ranges values between 2.1 and 2.4, in contrast with the general prescription, based on physical considerations of long range molecular interactions, that for intermolecular
interactions CSS+ COS = 2.10, 17, 52
3.2
Transferable SCS-MP2
modWhile the mp2modand scs-mp2mod protocols were originally devised to find the best set
of parameters for a specific homo-dimer, it is interesting to investigate to what extent the
scs-mp2mod method can be applied to a wider range of targets using the same basis set
and scaling factors for all the nine molecules here considered. Indeed, due to the limited
computational cost of the scs-mp2mod calculations, it would be greatly beneficial to
employ previously optimized basis sets, without performing any additional tuning through CCSD(T) reference data. Therefore, in this section, we give up system specific basis set
and scaling factors and check the possibility of using the same parameter set for a wide range of systems. To this end, a new reference CCSD(T)/CBS database, consisting in all dimer interaction energies with the exception of the more repulsive points, was employed
to obtain a unique set of scs-mp2mod parameters, i.e. five exponents for the 6-31G**
polarization functions and two SCS scaling factors, which can be in principle used for any pair of hetero-cyclic here considered. All parameters, as well as the RMSD and LUD indicators are reported in Table 5.
CSS COS αC αN αO αS αH RMSD LUD
2.061 0.018 0.274 0.481 0.348 0.230 0.183 0.199 0.707
Table 5: Orbital exponents of the polarization GTOs and scaling factors of the global scs-mp2mod
op-timization of the 6-31G** basis set. RMSD and LUD are in kcal/mol.
The exponents take reasonable values and also the scaling factors are similar to those obtained for the system specific optimizations. As expected, the statistical descriptors take less favorable values than the previous ones, although the accuracy is still satisfactory. In particular by looking at the data of Table 1 it is apparent that both the RMSD and
the LUD are still better than the system specific optimizations mp2mod , i.e. without
tuning the scaling factors.
To test the reliability of this global parameter set, a benchmark set of aromatic hetero-dimers was created by picking two different species among the compounds investigated in this work. Concretely, twelve different pairs were considered, namely Furan – Isoxazole, Furan – Oxazole, Thiophene – Isoxazole, Thiophene – Furan, Furan – Pyrimidine, Thio-phene – Pyridazine, Oxazole – Pyridine, Isoxazole – Pyrazine, Pyridine – Furan, Pyridine – Pyrimidine, Pyridine – Pyridazine and Pyrazine – Pyrimidine. For each dimer, two at-tractive geometries were considered, one in a stacked arrangement and one in a T-shaped one. All considered geometries are shown in detail in Figures L and M in the Supporting Information. The interaction energy of each of the resulting 24 conformers was
com-puted at CCSD(T)/CBS and scs-mp2mod level. Additionally, to better evaluate the gain
MP2 SCS-MP2mod
Set RMSD MAD LUD RMSD MAD LUD
two 5-membered rings 2.30 2.23 3.07 0.13 0.11 0.22
5-membered and 6-membered rings 2.16 2.04 2.78 0.23 0.06 0.52
two 6-membered rings 2.04 2.02 2.37 0.40 0.20 0.67
Stacked arrangements 2.58 2.57 3.07 0.32 0.14 0.67
T-shaped arrangements 1.68 1.63 1.95 0.17 0.07 0.39
all heterodimers 2.17 2.10 3.07 0.26 0.04 0.67
Table 6: Comparison, with respect of with reference CCSD(T)/CBS values, between standard MP2
calculations carried out with the 6-31G** basis set and scs-MP2mod
calculations performed with the modified 6-31G** basis set . The 24 dimer arrangements discussed in the text and shown in Figures L and M in the Supporting Information are considered as reference database. RMSD, MAD and LUD are all reported in kcal/mol.
all arrangements were also computed using the standard MP2 method with the original 6-31G** basis set. Results are reported in Figure 3 and summarized in Table 6.
Apart from the impressive improvement of the present results with respect to MP2/6-31G** calculations, we may notice that the accuracy is rather balanced within the several types of dimer and geometries. More in detail, the accuracy is well balanced among the hetero-dimers formed by either two 5-membered rings or one 5- and one 6-membered, whereas slightly less accurate prediction were obtained for the 6-membered rings hetero-dimers. As far as the geometrical arrangement is concerned, the performance is slightly better for the T-shaped than for the stacked arrangements, even if the limited number of dimer arrangements here considered does not allow any precise conclusion about the quality of the results as related to a specific geometry. Yet, this last test seems to confirm that the transferable basis set and scaling factors are able to deliver rather accurate estimates of the interaction energy. Considering the that choice of the pairs reported in Table 6 was random, a similar accuracy can reasonably be expected for any different pair of aromatic hetero-cycles at any geometry. This is of valuable importance in all situations where a relevant portion of the IPES should be explored for a wide range of molecular pairs.
F u ra n -I so xa zo le F u ra n -O xa zo le T h io p h e n e -I so xa zo le T h io p h e n e -F u ra n F u ra n -P yr im id in e T h io p h e n e -P yr id a zi n e O xa zo le -P yr id in e Iso xa zo le -P yr a zi n e F u ra n -P yr id in e P yr id in e -P yr im id in e P yr id in e -P yr id a zi n e P yr im id in e -P yr a zi n e
Figure 3: CCSD(T)/CBS (blueish bars), scs-mp2mod
(greenish) and MP2/6-31G** (reddish) interac-tion energies computed for the considered hetero-dimers in stacked (blue, green and red) and T-shaped (cyan, light green and orange) geometries (see Figure L and M in Supporting Information). Top: inter-action energies; bottom: relative error with respect to CCSD(T)/CBS. All energies are in kcal/mol.
4
Conclusions
In this paper we have presented a method to calculate the intermolecular energy of an important class of molecules. The main goal of this work is devising a cheap theoretical method, capable of delivering results at a level of accuracy sufficient to be profitably used in exploring large portions of the IPES. The computational convenience relies on joining the perturbative MP2 method with a modified basis set 6-31G**, with the further
inexpensive expedient of tuning the scaling factors of the the same-spin and opposite-spin components of the correlation energy. The main conclusion of the present effort is that
the optimization of these scaling factors in the scs-mp2mod has very relevant effects on
the accuracy of the results. The most accurate results were obtained for system specific optimization of the basis set and scaling factors, although in our opinion, even the use of a unique set of parameters for all the molecules here considered leads to satisfactory results. This generalization opens the way to the possibility of computing IPES for any
pair of molecules so enlarging the usefulness of scs-mp2mod to a rather large chemical
situations.
Supporting Information
Additional data and several details about the reported calculations not included
in this paper are available: dimer geometries, scs-mp2mod interaction energy curves,
CCSD(T)/scs-mp2mod correlation plots and starting and optimized scs-mp2mod
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